Limit Problem

[Richardnm]

The limit is lim(x->0) (tgx -sinx)/(x^3).
I can't get rid of the x on the denominator.
Detail, I can't use L'Hospital or any other derivative form to solve it, only factoring method...

If somebody could explain step by step...

thanks.

 

[Quaid]

The limit is lim(x->0) (tgx -sinx)/(x^3)

Do you know the meaning of tgx ?

I can't get rid of the x on the denominator.

Detail, I can't use L'Hospital or any other derivative form to solve it, only factoring method...

If somebody could explain step by step...

This is a tutoring web site; we do not -- in general -- provide step-by-step solutions.

Please read the forum guidelines. Here's a link to the summary page.

Ciao

 

[Richardnm]

Sorry, I guess I asked it wrong.

I do know what tangent is, and I know the trigonometric relations.
I started with (sinx/(x^3)*cosx))-sinx/(x^3). Separating the fundamental limit sinx/x from each equation, I get:
(1/(x^2)*cosx)) - (1/(x^3)).
And then I get stuck.

Is there another approach on the beginning or it's forward and I don't see it?

 

[Quaid]

I guess I asked it [wrongly].

I was confused by tgx because I do not recognize that symbol.

Who taught you to express the tangent of x as tgx ?

If we follow that pattern of notation, then the sine of x would be sex. :roll:

Please use standard notation. The tangent of x is tan(x).

I started with (sinx/(x^3)*cosx))-sinx/(x^3)

Separating the fundamental limit sinx/x from each equation, I get:
(1/(x^2)*cosx)) - (1/(x^3))

And then I get stuck.

Is there another approach?

I'm guessing that the exponent 3 (highlighted in red above) is a typographical error. (It should be 2.)

Hmmm, I'm not sure where you got tangent, from that last result, but let's continue from there.

cos(x)/x^2 - 1/x^2

As the denominators are the same, we can combine these ratios.

[cos(x) - 1]/x^2

Multiply top and bottom by cos(x)+1

In the numerator, you will have cos(x)^2-1. That can be expressed as -[1-cos(x)^2]

Use a basic identity, to write -[1-cos(x)^2] in terms of sine

Next, factor out [sin(x)/x]^2, and you will be able to evaluate the limit.

Cheers

 

[Richardnm]

Just got it, in the beginning I can multiply both numerator and denominator by cos(x) [Like that? Sorry about the english and the notation, I'm working on that... Thanks for the advices!]

After doing that, it goes nice and smooth

Thank you for your time...